It really depends on your priorities. Having worked involved with a few independent schools over the years- I would say one of the biggest differences is pastoral care.
Thinking about a parent that I knew, that had some anxiety issues & struggled to settle their child into nursery for a multitude of reasons. One very highly esteemed school wouldn't tolerate her need to settle in her child over time "because that wasn't their policy". She pulled her out after a few awful weeks. Which made trying again even more difficult for both of them.
The school she ended up going for, was more flexible on their policy because they recognised that was in the best need of the parent & child at that time. 5 years later, the child is positively thriving. Academically and emotionally, the parent is also much happier generally. The first school made her feel like a failure- just because of their policy (which wasn't really child friendly for ANY child to be very honest)
Now- with that example, it is not really just about the settling in policy. It is about the school's willingness to place the needs of the child (and parent) above the academic.
Another area where schools seem to differ greatly is creativity- I would look at the art work on the walls. Try & spot work that is completely child made, not adulterated or 'improved on' by adults. Look for individual differences. If you are in a school where it is a wall of very similar art work, that looks a little too perfect, that would set off alarm bells for me. How are children learning to develop their own ideas if they are being told exactly what to do, when & how? That is not creativity or even developmentally appropriate for nursery.
I would observe the staff- are they happy? Engaged? At child level? How do they interact with the children? Is it directive or more warm & friendly?
What about the children- how much choice do they get over their activities across the day? How much fun are they having? How structured & scheduled is their time? What is the balance of child led: adult led? How does play feature in their provision? (Thinking about opportunities for children to develop critical thinking, independence & problem solving. Play provides the more appropriate way to develop these skills at this age- but relies of the right environment and staff to be effective.)
Independent schools also have more flexibility on staff qualifications- so I would be asking about how many have early years training & experience.
Well that's just a few to start, again it really depends on your priorities. If you value only academics, your list might look quite different.